David Moyes remains unlikely to continue as West Ham’s manager despite finishing the season with a 3-1 victory over Everton at the London Stadium, with sources rating the Scot’s chances of being given a new contract at no more than 25%.

West Ham expect to reach a final decision at a board meeting next week and although Moyes has kept them in the Premier League, the east London club will consider a host of candidates in the coming days. If they prove attractive enough to one of their favoured picks, then Moyes is likely to be looking for a new job.

The same could be true of Sam Allardyce, who did not sound optimistic about seeing out the final year of his contract with Everton. “I can’t quite say I’m confident after all the rumours I’m reading because there is no smoke without fire, is there?” Allardyce said. “But we will wait and see when we meet up this week.”

Moyes was more positive. West Ham were in the bottom three when the 55-year-old replaced Slaven Bilic on a six-month deal in November and he is thought to be in line for a £1.65m bonus after leading them to 13th place. But while he has found a way to survive in awkward circumstances, he might not have impressed sufficiently to convince David Sullivan and David Gold, the owners, to reward him with a long-term deal.

West Ham are interested in Rafael Benítez, who might prove too expensive to prise away from Newcastle, and they have held informal talks with Manuel Pellegrini, although the former Manchester City manager has also been heavily linked with Sevilla. Other options include Burnley’s Sean Dyche, Huddersfield’s David Wagner and Marco Silva, the former Hull and Watford manager. Their preference is to hire a manager who has worked in the Premier League, although they have not denied making contact with Shakhtar Donetsk’s Paulo Fonseca last week.

They have also not completely ruled out appointing Moyes, who has misgivings of his own about taking the job. The former Everton manager hit out at the club’s culture of leaks after word of his row with Andy Carroll filtered through to the media earlier this month and he has challenged West Ham to show that they can match his ambition.

“We will meet next week,” Moyes said. “We’ve not arranged a date, not arranged a time. I don’t know if anybody can make that big of a difference in six months. Even Pep [Guardiola] took a bit of time to get this right. I don’t think anybody can really come and make a difference. You would need to give a bit longer. Everybody always thinks the grass is greener and there is something better out there. I would say, more often than not, it’s proved wrong.”

Moyes believes that he will receive offers from elsewhere if his time is up. “I’ll have options,” he said. “Look, it’s not us challenging each other. I’ve had a really good relationship with the board, I speak regularly with them. I would like to change things and maybe everybody is not happy with change.”

There is a feeling within West Ham’s squad that Moyes’s methods are old-fashioned but he provided a strong audition here. The hosts swept forward and took the lead in the 38th minute. Cheikhou Kouyaté’s pass ran beyond Marko Arnautovic but Michael Keane had pushed up, leaving a huge gap in the middle of Everton’s defence. Manuel Lanzini scampered on to the ball and used his left foot to beat Jordan Pickford with a precise shot from 18 yards.

Everton were bland for long spells and finishing eighth is unlikely to convince them to keep Allardyce, who hit out at suggestions that he has a troubled relationship with Wayne Rooney. “We’ve been very professional and very adult with our conversations,” he said. “You’ve been misled by somebody, or you have just decided to make it up.”

Rooney, who has an offer to join DC United, was absent with a knee injury and Everton toiled without him. West Ham doubled their lead midway through the second half when Arnautovic rolled away from Keane and caught out Pickford with a thunderous shot from long range. The Austrian has scored 11 times in the league under Moyes.

The visitors pulled a goal back when Oumar Niasse bundled the ball home from a corner with 16 minutes left, but West Ham sealed their first home win over Everton since 2007 when Pickford failed to get a strong enough hand to Lanzini’s curling shot.

Moyes ended with a light touch, joking that he can relax now that he is out of contract. “I’ll take my suit off at midnight, undo my tie a little bit, be back on the streets come midnight,” he said.